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Characterizations of Besov-Type and Triebel–Lizorkin–Type Spaces via Averages on Balls
Published online by Cambridge University Press: 20 November 2018
Abstract
Let $\ell \in \mathbb{N}$ and $\alpha \in (0,2\ell )$. In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaces via the sequence ${{\{f-{{B}_{\ell ,{{2}^{-k}}}}f\}}_{k}}$ consisting of the difference between $f$ and the ball average ${{B}_{\ell ,{{2}^{-k}}}}f$. These results lead to the introduction of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaceswith any positive smoothness order onmetricmeasure spaces. As special cases, the authors obtain a new characterization of Morrey–Sobolev spaces and ${{\text{Q}}_{\alpha }}$ spaces with $\alpha \in (0,1)$, which are of independent interest.
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- Copyright © Canadian Mathematical Society 2017
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